![]() |
Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > eqvrelsymb | Structured version Visualization version GIF version |
Description: An equivalence relation is symmetric. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised and distinct variable conditions removed by Peter Mazsa, 2-Jun-2019.) |
Ref | Expression |
---|---|
eqvrelsymb.1 | ⊢ (𝜑 → EqvRel 𝑅) |
Ref | Expression |
---|---|
eqvrelsymb | ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝐵𝑅𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqvrelsymb.1 | . . . 4 ⊢ (𝜑 → EqvRel 𝑅) | |
2 | 1 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → EqvRel 𝑅) |
3 | simpr 488 | . . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → 𝐴𝑅𝐵) | |
4 | 2, 3 | eqvrelsym 36000 | . 2 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → 𝐵𝑅𝐴) |
5 | 1 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝐵𝑅𝐴) → EqvRel 𝑅) |
6 | simpr 488 | . . 3 ⊢ ((𝜑 ∧ 𝐵𝑅𝐴) → 𝐵𝑅𝐴) | |
7 | 5, 6 | eqvrelsym 36000 | . 2 ⊢ ((𝜑 ∧ 𝐵𝑅𝐴) → 𝐴𝑅𝐵) |
8 | 4, 7 | impbida 800 | 1 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝐵𝑅𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 class class class wbr 5030 EqvRel weqvrel 35630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-refrel 35912 df-symrel 35940 df-trrel 35970 df-eqvrel 35980 |
This theorem is referenced by: eqvrelth 36006 |
Copyright terms: Public domain | W3C validator |